Series converge?

**xinglongdada** · October 24th 2012, 04:46 AM

Let $a_n>0$ , $\lim_{n\to\infty}a_n=a>0$ . Study the converegence or divergence of the series $\sum_{n=1}^\infty (a/a_n)^n$ .

Thanks.

**Prove It** · October 24th 2012, 04:50 AM

Originally Posted by xinglongdada

Let $a_n>0$ , $\lim_{n\to\infty}a_n=a>0$ . Study the converegence or divergence of the series $\sum_{n=1}^\infty (a/a_n)^n$ .

Thanks.

I think this series must diverge. If $\displaystyle \begin{align*} a_n \to a \end{align*}$ then $\displaystyle \begin{align*} \frac{a}{a_n} \to \frac{a}{a} = 1 \neq 0 \end{align*}$ .

**xinglongdada** · October 24th 2012, 05:00 AM

But we may have the limit $1^\infty$ .

**xinglongdada** · October 24th 2012, 05:02 AM

Originally Posted by Prove It

I think this series must diverge. If $\displaystyle \begin{align*} a_n \to a \end{align*}$ then $\displaystyle \begin{align*} \frac{a}{a_n} \to \frac{a}{a} = 1 \neq 0 \end{align*}$ .

For example,
$\lim_{n\to\infty}(1-1/\sqrt{n})^n=\lim_{n\to\infty}e^{-\sqrt{n}}=0.$

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